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25.    (a)   Give an example of a sequence (/„) of continuous real-valued functions on an
interval I c R which converges simply at each point of I but is such that there exists
no nonempty open interval contained in I on which the sequence converges uniformly
(cf. (12.7.1)).

(b) Let (/„) be a sequence of complex analytic functions on an open set D c C
which converges simply at each point of D. Show that there exists a dense open set U
in D such that, for each z0 e U, there exists a neighborhood of z0 on which the
sequence (/„) is uniformly convergent. (Consider the sequence of functions zh-» \fn(z)\
and use (9.13.2).)

26.    Let t\-^A(t) be a continuous mapping of the interval R+ = [0, + oo[ of R into the
space of real n x n matrices, and let t*-+C(t) be the solution of the homogeneous
linear differential equation U' = A(t)U which is equal to the unit matrix when t = 0
(10.8.4). Let EI denote the vector subspace of Rn consisting of vectors y such that the
function t\-~+C(t) - y is bounded on [0, + oo[, and let E2 be a supplement of EX in
R". Let PI : R" -* Ex and P2 : Rn -> E2 be the projections associated with the direct-
sum decomposition R" = Ex © E2 (5.4).

(a) Suppose that, for each function fe <tf£n(R+), the differential equation
Kr — A(t) - x +/(0 has at least one solution which is bounded on R+ . Show that, for
each function / e ^°n (R+), there exists a unique vector y(f) e E2 such that the
unique solution uf of the equation x' = A(t) - x + f(t) which takes the value y(f) at
the point 0 is bounded in R+ , and that there exists a constant c > 0 such that

(Consider the space F of continuously differentiate functions x on R+ with values
in R" such that x and x' — A • x are bounded on R+ and #(0) e E2. Show that the

is a norm on F, with respect to which F is a Banach space. Then apply Banach's
theorem to the linear mapping jti—>x' — A -xofF into #£, (R+), bearing in mind the
definition of E2.)

(b) Deduce from (a) that a necessary and sufficient condition for the equation
x' = A(t) - x+f(t) to have a solution which is bounded on R+, for all functions

sup ( f \\C(t)P± C"1(s)\\ ds + f     ||C(t)P2C'l(s) || ds] < + oo

t>o \Jo                                  J r                                 /

(cf. (10.8.6) and (13.14.4)).

27. Let/: [a,£>[~»R be an increasing function, continuous on the right. Suppose that
the countable set of points of discontinuity of / is dense in [a, b[ (Section 3.15,
Problem). Show that the set of points x e [a, b[ at which /has a finite derivative on the
right is meager. (For each integer n > 0, let An be the set of points x e [a, b[ such
that there exist points y, z e [a, b[ satisfying x < y < z < x + l//zand

/Cy)-/(*)    /(*)-/(*)

y — x              z — x

Show that AM is a dense open subset of [a, b[.)t, for each sequence (£„) of real